# University of Toronto's Symplectic Geometry Seminar

Oct. 28 2002, 2:10pm

SS5017A

## Andrzej Derdzinski

###
Ohio State University

##
"Four-dimensional spin^c-manifolds with Higgs fields"
(joint work with T. Januszkiewicz)

** Abstract: **
We define a `Higgs field' for a four-dimensional spin^c-manifold to be
a smooth section of its positive half-spinor bundle, transverse to the zero
section, and defined only up to a positive functional factor. This is intended
to be a generalization of almost complex structures on real four-manifolds,
each of which may in fact be treated as a Higgs field without zeros for
a specific spin^c-structure. The notions of totally real or pseudoholomorphic
immersions of real surfaces in an almost complex manifold of real dimension
four have straighforward generalizations to the case of a spin^c-manifold with
a Higgs field.
Since a spin^c-structure with a Higgs field exists on every orientable
four-manifold, immersions of surfaces that are pseudoholomorphic relative to
a Higgs field might conceivably be used as probing tools to study the topology
of a large class of manifolds in dimension four, in analogy with Taubes's work
in the symplectic case.
Our results consist, first, in showing that totally real immersions of closed
oriented surfaces in four-dimensional spin^c-manifolds with Higgs fields have,
basically, the same properties as in the almost-complex case, and, secondly,
in providing a description of all pseudoholomorphic immersions of
such
surfaces in the four-sphere endowed with a "standard" Higgs field.